Limit cycles for dynamic crawling locomotors with periodic prescribed shape

We study the asymptotic evolution of a family of dynamic models of crawling locomotion, with the aim to introduce a well-posed characterization of a gait as a limit behaviour. The locomotors, which might have a discrete or continuous body, move on a line with a periodic prescribed shape change, and might possibly be subject to external forcing (e.g. crawling on a slope). We discuss how their behaviour is affected by different types of friction forces, including also set-valued ones such as dry friction. We show that, under mild natural assumptions, the dynamics always converge to a relative periodic solution. The asymptotic average velocity of the crawler yet might still depend on its initial state, so we provide additional assumption for its uniqueness. In particular, we show that the asymptotic average velocity is unique both for strictly monotone friction forces, and also for dry friction, provided in the latter case that the actuation is sufficiently smooth (for discrete models) or that the friction coefficients are always nonzero (for continuous models). We present several examples and counterexamples illustrating the necessity of our assumptions.